Modeling the spread of COVID-19 as a consequence of undocumented immigration toward the reduction of daily hospitalization: Case reports from Thailand

At present, a large number of people worldwide have been infected by coronavirus 2019 (COVID-19). When the outbreak of the COVID-19 pandemic begins in a country, its impact is disastrous to both the country and its neighbors. In early 2020, the spread of COVID-19 was associated with global aviation. More recently, COVID-19 infections due to illegal or undocumented immigration have played a significant role in spreading the disease in Southeast Asia countries. Therefore, the spread of COVID-19 of all countries’ border should be curbed. Many countries closed their borders to all nations, causing an unprecedented decline in global travel, especially cross-border travel. This restriction affects social and economic trade-offs. Therefore, immigration policies are essential to control the COVID-19 pandemic. To understand and simulate the spread of the disease under different immigration conditions, we developed a novel mathematical model called the Legal immigration and Undocumented immigration from natural borders for Susceptible-Infected-Hospitalized and Recovered people (LUSIHR). The purpose of the model was to simulate the number of infected people under various policies, including uncontrolled, fully controlled, and partially controlled countries. The infection rate was parameterized using the collected data from the Department of Disease Control, Ministry of Public Health, Thailand. We demonstrated that the model possesses nonnegative solutions for favorable initial conditions. The analysis of numerical experiments showed that we could control the virus spread and maintain the number of infected people by increasing the control rate of undocumented immigration across the unprotected natural borders. Next, the obtained parameters were used to visualize the effect of the control rate on immigration at the natural border. Overall, the model was well-suited to explaining and building the simulation. The parameters were used to simulate the trends in the number of people infected from COVID-19.

The data we used were described as follows: i) Hospitalized: this is the daily hospitalized people, and we used this data for fitting parameters and simulation as an actual data. ii) Total hospitalized: this data is the cumulative number of hospitalized people because the report from the government is the number of cases that treat in the hospital. iii) Recovered: this is the cumulative number of recovered people, and we used this data to calculate the recovery rate of the hospitalized. iv) Death: this is the cumulative number of deaths, and we used this data to calculate the death rate of the hospitalized.
Appendix A2: The proof of model positivity For the LUSIHR model, to prove that their solutions were non-negative (all populations more than or equal to zero) if the initial conditions were non-negative, we can apply the concept of the proof the model positivity of the feedback vaccination law for SIR model [19]. The population functions ( ) L t , ( ) U t , ( ) S t , ( ) I t , ( ) H t , and ( ) R t are continuous and differentiable as shown in Equations Error! Reference source not found. -Error! Reference source not found. in the main manuscript. Then, we need to show that if the population decreased to zero, the derivative was positive to increase the population to be positive again.
Let us consider each equation in the proposed system of equations as the following: For the LUSIHR model that 0 1    , Equation Error! Reference source not found. is for legal immigration ( ) L t and Let the initial condition of the population be positive, (0) 0 L  . For some l t that ( ) 0,  is the recruited rate that must be greater than zero. As the derivative of L for l t is positive, ( ) 0 Equation Error! Reference source not found. is for undocumented immigration over the natural border ( ) U t and 2 3 4 Let the initial condition of the population be positive, (0) 0  is the recruited rate and (1 )   is between 0 and 1.
As the derivative of U related to u t is positive, ( ) 0 Equation Error! Reference source not found. is for susceptible population ( ) S t and Let the initial condition of the population be positive, (0) 0 Equation Error! Reference source not found. is for infected population ( ) I t and Let the initial condition of the population be positive, (0) 0 Equation Error! Reference source not found. is for the hospitalized population ( ) H t and Let the initial condition of the population be positive, (0 Let the initial condition of the population be positive, (0) 0 For the LSIHR model where 1   , the positivity of ( ) L t , ( ) H t and ( ) R t is shown in the LUSIHR model. So, we will prove only ( ) S t and ( ) I t .
Equation Error! Reference source not found. is for susceptible population ( ) S t and Let the initial condition of the population be positive, (0) 0 Equation Error! Reference source not found. is for the infected population ( ) I t and Let the initial condition of the population be positive, (0) 0 In conclusion, from the above analysis, all the solutions are non-negative for all times t and these proposed models can be applied to analyze its equilibrium points further and simulate and predict the growth of the populations.
Appendix A3: The analysis of an equilibrium point For the LUSIHR model, the equilibrium point was calculated by setting Equations Error! Reference source not found. -Error! Reference source not found. equal to zero as in the following non-linear system: The solution of the system of Equations (1) -( 6 ) was an equilibrium point. We obtained only one EE point in which the infection was not removed from the system; this is shown as the following.
By Equation (1): and denote 1 1 By Equation (3): Substituting Equation (7) in the previous equation, we obtain We will show that the condition will always be 2 2 1 1 4 Notice that 2 4 0 C AB    . Therefore, we take a power of two and then we get We rearrange the above expression, and we get which is the same inequality as the above situation. Thus, it is a contradiction and we can conclude that 2 2 1 4 Therefore, we have only one equilibrium point that satisfies the system of the equation as 2 * 4 2 By Equation (5): By Equation (6): The EE point of the LUSIHR model is For the LSIHR model, the equilibrium points were calculated by using Equations (1), (5) -(6) and setting the differential Equations Error! Reference source not found. -Error! Reference source not found. as zero. Therefore, we obtained The solutions for Equations (1), ( 5 ) -( 6 ) , and (8) -(9) represented an equilibrium point. We obtained two equilibrium points, one of which was the EE point and the other a DFE point as follows: By Equation (1): By Equation (8): By Equation (5): By Equation (6): In the above expression, there is no positive real root using Descartes' rule of signs if 0 p  and 0 q  . Therefore, the condition that makes 0 p  and 0 q  is In the same way with the EE, we obtain   Table A5 (below) shows the maximum number (peaks) of hospitalized people from the simulation results under the first and the second waves. The result shows that when the control rate of the undocumented immigrants reached below 50%, the number of cases in the second peak is expected to rise by at least 60% increase in the number of the patient and if the undocumented immigrants were controlled by 80%, the impact of COVID-19 would be reduced and the number of new cases in the second wave would rise lower than 50% compared to that of the first wave.